Trigonometric Ratios in Right Triangles

Trigonometric Ratios are based on the Concept of Similar Triangles!

All 45º- 45º- 90º Triangles are Similar!

45 º

2

2

22

45 º

1

1

2

45 º

1

2

1

2

1

All 30º- 60º- 90º Triangles are Similar!

1

60º

30º

½

23

32

60º

30º

2

4

2

60º

30º

1

3

All 30º- 60º- 90º Triangles are Similar!

10 60º

30º

5

35

2 60º

30º1

3

160º

30º 21

23

A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legs of the triangle.

cb

a

90

Naming Sides of Right Triangles

The Tangent Ratio

Tangent Tangent Adjacent

Opposite

There are a total of six ratios that can be madeThere are a total of six ratios that can be madewith the three sides. Each has a specific name.with the three sides. Each has a specific name.

The Six Trigonometric Ratios(The SOHCAHTOA model)

Adjacent

OppositeTangentθ

Hypotenuse

AdjacentCosineθ

Hypotenuse

OppositeSineθ

The Six Trigonometric Ratios

Adjacent

OppositeTangentθ

Hypotenuse

AdjacentCosineθ

Hypotenuse

OppositeSineθ

Opposite

AdjacentCotangentθ

Adjacent

HypotenuseSecantθ

Opposite

HypotenuseCosecantθ

The Cosecant, Secant, and Cotangent of The Cosecant, Secant, and Cotangent of are the Reciprocals of are the Reciprocals of

the Sine, Cosine,and Tangent of the Sine, Cosine,and Tangent of

Reciprocal Identities

Quotient Identities

Find the value of each of the six trigonometric functions of the angle

Adjacent

12 13

c = Hypotenuse = 13

b = Opposite = 12

a

b

c

Adjacent = 5

Opposite =

Hypotenuse =

12

13

25

70

h

h = 23.49

Solving a Problem withthe Tangent Ratio

60º

53 ft

h = ?

We know the angle and the We know the angle and the side adjacent to 60º. We want to side adjacent to 60º. We want to know the opposite side. Use theknow the opposite side. Use thetangent ratio:tangent ratio:

ft 92353

531

3

5360tan

h

h

h

adj

opp

1

2 3

Why?